Where are the hard
manipulation problems?
Toby Walsh
NICTA and UNSW
Sydney
Australia
Escaping Gibbard-Sattertwhaite
 Complexity may be a
barrier to manipulation?
 Some voting rules (like
STV) are NP-hard to
manipulate
[Bartholdi, Tovey & Trick 89, Bartholdi
& Orlin 91]
Escaping Gibbard-Sattertwhaite
 Complexity may be a
barrier to manipulation?
 Some voting rules (like
STV) are NP-hard to
manipulate
NP-hardness = as hard as SAT,
TSP, … and other NPcomplete problems
Best known complete algorithm
takes exponential time
Start of intractability
Complexity as a friend?
 NP-hardness is only
worst case
 Manipulation might be
easy in practice
Hardness of manipulation
in practice?


Theoretical tools

Average case

Approximability
Empirical tools

Heuristic methods

Phase transition (cf other NP-hard
problems like SAT and TSP)
Hardness of manipulation
in practice?


Theoretical tools

Average case

Approximability
Empirical tools

Heuristic methods

Phase transition (cf other NP-hard
problems like SAT and TSP)
Veto rule

Simple rule to analyse



Each voter gets one veto
Candidate with least vetoes wins
But on border of complexity


NP-hard to manipulate
constructively with 3 or more
candidates, weighted votes
Polynomial to manipulate
destructively
]
Manipulating veto rule
Manipulation not possible
with 2 candidates


If the coalition want A to
win then veto B
Manipulating veto rule
Manipulation possible with
3 candidates


Voting strategically can
improve the result
Manipulating veto rule
Suppose




A has 4 vetoes
B has 2 vetoes
C has 3 vetoes
Coalition of 5 voters


Prefer A to B to C
Manipulating veto rule
Suppose




A has 4 vetoes
B has 2 vetoes
C has 3 vetoes
Coalition of 5 voters



Prefer A to B to C
If they all veto C, then B
wins
Manipulating veto rule
Suppose




A has 4 vetoes
B has 2 vetoes
C has 3 vetoes
Coalition of 5 voters



Prefer A to B to C
Strategic vote is for 3 to veto
B and 2 to veto C
Manipulating veto rule
With 3 or more candidates
Unweighted votes



Manipulation is polynomial to
compute
Weighted votes


Destructive manipulation is
polynomial
Constructive manipulation is NPhard (=number partitioning)
Uniform votes
n agents
3 candidates
coalition of size m
weights from [0,k]

Weighted form of impartial culture model
Phase transition
Phase transition
Phase transition
Prob = 1- 2/3e-m/n
Phase transition
Phase transition

Same result with other distributions of votes
 Different size weights
 Normally distributed weights
 ..
Hung elections


n voters have vetoed one
candidate
coalition of size m has twice
weight of these n voters
Hung elections


n voters have vetoed one
candidate
coalition of size m has twice
weight of these n voters
Hung elections



n voters have vetoed one
candidate
coalition of size m has twice
weight of these n voters
But one random voter with
enough weight makes it easy
What if votes are unweighted?
 STV is then one of the
most difficult rules to
manipulate
 One of few rules where
it is NP-hard
 Multiple rounds,
complex manipulations
...
STV phase transition
Varying number of candidates
STV phase transition
 Smooth not sharp?
 Other smooth transitions: 2-COL, 1in2-SAT, …
STV phase transition
Fits 1.008m with coefficient of determination R2=0.95
STV phase transition
Varying number of voters
STV phase transition
Varying number of agents
STV phase transition
 Similar results with many voting distributions
 Uniform votes (IC model)
 Single-peaked votes
 Polya-Eggenberger urn model (correlated votes)
 Real elections …
Correlated votes
Polya-Eggenberger model (50% chance 2nd vote=1st vote,..)
Sampling real elections
 NASA Mariner space-craft experiments
 32 candidate trajectories, 10 scientific teams
 UCI faculty hiring committee
 3 candidates, 10 votes
Sampling real elections
 Fewer candidates
 Delete candidates
randomly
 Fewer voters
 Delete voters randomly
 More candidates
 Replicate, break ties
randomly
 More voters
 Sample real votes with
given frequency
NASA phase transition
Coalitions
Coalitions
Conclusions
 In many cases, NP hardness does not appear to be a
barrier to manipulation!
 How else might we escape GS?
 Higher complexity classes
 Undecidability
 Incentive mechanisms (money)
 Cryptography (one way functions)
 Uncertainty (random voting methods)
 Quantum
…
Questions?
Background reading
[T. Walsh, Where are the really hard manipulation
problems? The phase transition in manipulating the
veto rule, Proc. of IJCAI 2009]
[T. Walsh, An Empirical Study of the Manipulability
of Single Transferable Voting, Proc. of ECAI 2010]